Efficient Non-Uniform Quantum Hermite Transform through Adaptive Sampling

Abstract:On the span of the first $N$ oscillator modes, Gauss--Hermite quadrature gives an exact change of basis between mode coefficients and $N$ weighted position space samples. We implement this transform with $O(N\operatorname{polylog}(N,1/\varepsilon))$ logical gates and polylogarithmic quantum width. The operator-error bound $\varepsilon$ holds on arbitrary superpositions and includes all auxiliary registers. The construction uses signed averages on adaptive windows to convert uniform-grid samples into weighted Hermite-root samples. Their varying widths control the amplification cost, giving the near-linear bound.

Quantum Physics (quant-ph); Distributed, Parallel, and Cluster Computing (cs.DC); Emerging Technologies (cs.ET)

arXiv:2609.20739 [quant-ph]

(or arXiv:2609.20739v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.20739 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

cat_coefficients.npy

coherent_coefficients.npy

frft_density.npy

frft_evolved_coefficients.npy

hermite_demonstrations.py

hermite_demonstrations_validation.json

noise_populations.csv

noise_simulation.py

noise_validation.json

qho_classical_positions.npy

qho_evolved_coefficients.npy

requirements.txt

test_tail_squared.py

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